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THEORY OF ELASTICITY
Corresponding Member of the Academy of Sciences of the USSR E. I. GRIGOLYUK, P. P. CHULKOV
ON THE LINEAR THEORY OF THREE-LAYER SHELLS WITH A RIGID CORE
The general equations of equilibrium and free vibrations of thin three-layer shells of asymmetric construction with a rigid core compressed in the transverse direction are considered. In deriving the equations, the following assumptions are made. The load-bearing layers and the core have constant thicknesses and are made of different orthotropic materials. The core is regarded as a three-dimensional body, and it is assumed that the displacements of its points are approximated with sufficient accuracy by linear functions of the transverse coordinate. For the load-bearing layers the Kirchhoff—Love hypotheses are adopted. In contrast to the theory of shallow shells \((^{1,2})\), as well as in \((^3)\), refined expressions are adopted for the angles of inclination of the normal to the original surface of the shell. It is assumed that the deformations remain elastic at all times.
The equations of motion are obtained from the variation of the total energy of the shell. In calculating the variation of the potential energy of the core, along with the energy of transverse shear, the energy of transverse compression was taken into account.
Taking as the original surface the middle surface of the core, we introduce on it coordinates \(x_1, x_2\), referred to the lines of principal curvatures. The coordinate referred to the outer normal will be denoted by \(z\). Let the lower index of a component \(i = 1, 2\) indicate that the given component belongs to the \(i\)-th direction, and the upper index \(j = 1, 2, 3\) to the upper layer, the lower layer, and the core, respectively. Partial differentiation with respect to \(x_i\) will be denoted by the lower index \(i\) after a comma.
We denote by \(h_1, h_2, 2c\) and \(\rho_1, \rho_2, \rho\) the thicknesses and specific densities, respectively, of the first and second load-bearing layers and of the core; by \(A_i\) and \(k_i\) the Lamé coefficients and the principal curvatures of the original surface; by \(H_i = A_i(1 + k_i z)\) the Lamé coefficients of a surface equidistant from the original one; by \(w\) and \(u_i\) the normal and tangential displacements of points of the original surface; by \(2a_i\) the absolute shear of the boundary surfaces of the core along the \(x_i\) axis; by \(v\) the absolute approach of the original and lower surfaces of the core; by \(q_i\) the tangential surface load; by \(p^+\), \(p^-\) the normal surface loads applied respectively to the outer and inner sides of the composite shell; and by \(t\) the time.
The displacements will be written in the following form:
\[ w^z = \begin{cases} w + v, & c \leq z \leq c + h_1,\\ w + \dfrac{z}{c}v, & -c \leq z \leq c,\\ w - v, & -c - h_2 \leq z \leq -c; \end{cases} \tag{1} \]
\[ u_i^z = \begin{cases} u_i + a_i + (z - c)\left[(u_i + a_i)k_i - H_i^{-1}(w_{,i} + v_{,i})\right], & c \leq z \leq c + h_1,\\ u_i + \dfrac{z}{c}a_i, & -c \leq z \leq c,\\ u_i - a_i + (z + c)\left[(u_i - a_i)k_i - H_i^{-1}(w_{,i} - v_{,i})\right], & -h_2 - c \leq z \leq -c. \end{cases} \tag{2} \]
We introduce the specific forces and moments as follows:
\[ \begin{gathered} T_{ij}=T_{ij}^{1}+T_{ij}^{2}+T_{ij}^{3};\qquad M_{ij}^{+}=M_{ij}^{1}+M_{ij}^{2};\\ M_{ij}^{-}=M_{ij}^{1}-M_{ij}^{2};\qquad \widetilde{M}_{ij}=M_{ij}^{3}+c\,(T_{ij}^{1}-T_{ij}^{2}); \end{gathered} \tag{3} \]
\[ \begin{gathered} T_{ij}^{1}=\int_{c}^{c+h_{1}}\sigma_{ij}^{1}\,dz;\qquad T_{ij}^{2}=\int_{-c-h_{2}}^{-c}\sigma_{ij}^{2}\,dz;\qquad T_{ij}^{3}=\int_{-c}^{+c}\sigma_{ij}^{3}\,dz;\\ M_{ij}^{1}=\int_{c}^{c+h_{1}}\sigma_{ij}^{1}(z-c)\,dz;\qquad M_{ij}^{2}=\int_{-c-h_{2}}^{-c}\sigma_{ij}^{2}(z+c)\,dz;\qquad M_{ij}^{3}=\int_{c-}^{+c}\sigma_{ij}^{3}z\,dz; \end{gathered} \tag{4} \]
\[ T_{3}=\int_{-c}^{c}\sigma_{3}\,dz;\qquad Q_{1}=\int_{-c}^{c}\sigma_{13}\,dz;\qquad Q_{2}=\int_{-c}^{c}\sigma_{23}\,dz. \]
The equations of motion have the form:
\[ \begin{gathered} (A_{2}T_{11})_{,1}-A_{2,1}T_{22}+A_{1}^{-1}(A_{1}^{2}T_{12})_{,2} +k_{1}(A_{2}M_{11}^{+})_{,1}-\\ {}-k_{1}A_{2,1}M_{22}^{+}+k_{1}A_{1}^{-1}(A_{1}^{2}M_{12}^{+})_{,2} +(A_{1}k_{1}M_{12}^{+})_{,2}+\\ {}+k_{2}A_{1,2}M_{12}^{+}+k_{1}A_{1}A_{2}Q_{1} =-A_{1}A_{2}\left(q_{1}-\Omega_{11}\frac{\partial^{2}u_{1}}{\partial t^{2}}\right); \\[0.8em] -A_{1,2}T_{11}+(A_{1}T_{22})_{,2}+A_{2}^{-1}(A_{2}^{2}T_{12})_{,1} -k_{2}A_{1,2}M_{11}^{+}+\\ {}+k_{2}(A_{1}M_{22}^{+})_{,2} +A_{2}^{-1}(A_{2}^{2}M_{12}^{+})_{,2} +(A_{2}k_{2}M_{12}^{+})_{,1}+\\ {}+k_{1}A_{2,1}M_{12}^{+}+k_{2}A_{1}A_{2}Q_{2} =-A_{1}A_{2}\left(q_{2}-\Omega_{11}\frac{\partial^{2}u_{2}}{\partial z^{2}}\right); \\[0.8em] (A_{2}\widetilde{M}_{11})_{,1}-A_{2,1}\widetilde{M}_{22} +A_{1}^{-1}(A_{1}^{2}\widetilde{M}_{12})_{,2} +(A_{1}k_{1}cM_{11}^{-})_{,1} -k_{1}cA_{2,1}M_{22}^{-}+\\ {}+k_{1}cA_{1}^{-1}(A_{1}^{2}M_{12}^{-})_{,2} +(A_{1}k_{1}cM_{12}^{-})_{,2} +k_{2}cA_{1,2}M_{12}^{-} -A_{1}A_{2}Q_{1}=0; \\[0.8em] -A_{1,2}\widetilde{M}_{11}+(A_{1}\widetilde{M}_{22})_{,2} +A_{2}^{-1}(A_{2}^{2}\widetilde{M}_{12})_{,1} -k_{1}cA_{1,2}M_{11}^{-} -k_{1}c(A_{1}M_{22}^{-})_{,2}+\\ {}+k_{1}cA_{2}^{-1}(A_{2}^{2}M_{12}^{-})_{,1} +(A_{2}k_{2}cM_{12}^{-})_{,1} +k_{1}cA_{2,1}M_{12}^{-} -A_{1}A_{2}Q_{2}=0; \tag{5} \\[0.8em] A_{1}A_{2}(k_{1}T_{11}+k_{2}T_{22}) -\bigl(A_{1}^{-1}(A_{2}M_{11}^{+})_{,1}\bigr)_{,1} +\bigl(A_{2}^{-1}A_{1,2}M_{11}^{+}\bigr)_{,2}-\\ {}-\bigl(A_{2}^{-1}(A_{1}M_{22}^{+})_{,2}\bigr)_{,2} +\bigl(A_{1}^{-1}A_{2,1}M_{22}^{+}\bigr)_{,1} -\bigl(A_{2}^{-2}(A_{2}^{2}M_{12}^{+})_{,1}\bigr)_{,2}-\\ {}-\bigl(A_{1}^{-2}(A_{1}^{2}M_{12}^{+})_{,2}\bigr)_{,1} -(A_{2}Q_{1})_{,1}-(A_{1}Q_{2})_{,2}=\\ {}=A_{1}A_{2}\left[(p^{+}+p^{-})-\Omega_{11}\frac{\partial^{2}w}{\partial z^{2}} +\Omega_{12}\frac{\partial^{2}v}{\partial z^{2}}\right]; \\[0.8em] A_{1}A_{2}c^{-1}(k_{1}\widetilde{M}_{11}+k_{2}\widetilde{M}_{22}) +A_{1}A_{2}c^{-1}T_{3} -\bigl(A_{1}^{-1}(A_{2}M_{11}^{-})_{,1}\bigr)_{,1}+\\ {}+\bigl(A_{2}^{-1}A_{1,2}M_{11}^{-}\bigr)_{,2} -\bigl(A_{2}^{-1}(A_{1}M_{22}^{-})_{,2}\bigr)_{,2} +\bigl(A_{1}^{-1}A_{2,1}M_{22}^{-}\bigr)_{,1}-\\ {}-\bigl(A_{2}^{-2}(A_{2}^{2}M_{12}^{-})_{,1}\bigr)_{,2} -\bigl(A_{1}^{-2}(A_{1}^{2}M_{12}^{-})_{,2}\bigr)_{,1}-\\ {}-\frac{2}{3}c\left[G_{2}(A_{2}v_{,1}A_{1}^{-1})_{,1} +G_{3}(A_{1}v_{,2}A_{2}^{-1})_{,2}\right]=\\ {}=A_{1}A_{2}\left[(p^{+}-p^{-})+\Omega_{12}\frac{\partial^{2}w}{\partial z^{2}} +\Omega_{22}\frac{\partial^{2}v}{\partial z^{2}}\right]. \end{gathered} \]
The boundary conditions are determined from
\[ \begin{aligned} \int_{\alpha_1}^{\alpha_2} \Biggl\{& (T_{22}+k_2M_{22}^{+})\,\delta u_2 +(T_{12}+2k_1M_{12}^{+})\,\delta u_1 +(\widetilde{M}_{22}+k_2 c M_{22}^{-})\,\frac{\delta a_2}{c} \\ &+(\widetilde{M}_{12}+2k_1 c M_{12}^{-})\,\frac{\delta a_1}{c} +(A_1A_2)^{-1}\bigl[(A_1M_{22}^{+})_{,2}-A_{1,2}M_{11}^{+} \\ &\qquad\qquad +2(A_2M_{12}^{+})_{,1}+A_1A_2Q_2\bigr]\,\delta w +(A_1A_2)^{-1}\bigl[(A_1M_{22}^{-})_{,2} \\ &\qquad\qquad -A_{1,2}M_{11}^{-}+2(A_2M_{12}^{-})_{,1} +\tfrac{2}{3}cG_3A_1v_{,2}\bigr]\,\delta v \\ &\qquad\qquad -M_{22}^{+}A_2^{-1}\delta w_{,2} -M_{22}^{-}A_2^{-1}\delta v_{,2} \Biggr\} A_1\,dx_1\Big|_{\beta_1}^{\beta_2} \\ &\qquad -2M_{12}^{+}\delta w\Big|_{\alpha_1\beta_1}^{\alpha_2\beta_2} -2M_{12}^{-}\delta v\Big|_{\alpha_1\beta_1}^{\alpha_2\beta_2}; \end{aligned} \tag{6} \]
the integral corresponding to the other edge of the shell is obtained from (6) by the substitutions \(\alpha_1 \leftrightarrow \beta_1\), \(\alpha_2 \leftrightarrow \beta_2\), \(1 \leftrightarrow 2\), \(G_3 \to G_2\). Here \(G_2\) and \(G_3\) are the shear moduli of the core material, respectively for the planes \(x_1oz\) and \(x_2oz\); \(\Omega_{11}=\rho_1h_1+\rho_2h_2+2\rho c\); \(\Omega_{22}=\rho_1h_1+\rho_2h_2+\tfrac{2}{3}\rho c\); \(\Omega_{12}=\rho_1h_1-\rho_2h_2\).
We note that the system of equations presented is symmetric.
Institute of Hydrodynamics
Siberian Branch of the Academy of Sciences of the USSR
Received
20 XI 1962
CITED LITERATURE
- Kh. M. Mushtari, Izv. AN SSSR, OTN, Mekh. i mashinostr., No. 2 (1961).
- E. I. Grigolyuk, Izv. AN SSSR, OTN, No. 1 (1958).
- E. I. Grigolyuk, Yu. P. Kiryukhin, Izv. Sibirsk. otd. AN SSSR, No. 3 (1962).